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On the role of the degradation rates forthe dominance of the fittest in Eigen'squasispecies modelTanya Kostova a , Carol Zhou b & Adam Zemla ba National Science Foundation, 4201 Wilson Boulevard, suite 1025,Arlington, VA, 22230, USAb Lawrence Livermore National Laboratory, 7000 East Avenue,Livermore, CA, 94550, USAVersion of record first published: 20 Apr 2011.

To cite this article: Tanya Kostova , Carol Zhou & Adam Zemla (2011): On the role of thedegradation rates for the dominance of the fittest in Eigen's quasispecies model, Journal ofBiological Dynamics, 5:5, 531-548

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Journal of Biological DynamicsVol. 5, No. 5, September 2011, 531–548

On the role of the degradation rates for the dominance of thefittest in Eigen’s quasispecies model

Tanya Kostovaa*, Carol Zhoub and Adam Zemlab

aNational Science Foundation, 4201 Wilson Boulevard, suite 1025, Arlington, VA 22230, USA;bLawrence Livermore National Laboratory, 7000 East Avenue, Livermore, CA 94550, USA

(Received 28 September 2010; final version received 26 November 2010 )

The existence of an error threshold of the mutation rate in Eigen’s quasispecies model has been computa-tionally demonstrated exclusively in the case when the degradation rates of all model genotypes are equal.Here we explore the case with different degradation rates and demonstrate examples for which the typethat has highest fitness in the absence of mutation can preserve its dominance independently of the valueof the mutation rate. The examples are formulated based on analysis of the equilibria at the two extrememutation rate values and suggest absence of an error threshold in a number of cases, most prominentlywhen the degradation rate of the wild type is much smaller that the degradation rates of the mutants.

Keywords: quasispecies; viral evolution; error threshold; genotype dominance; spectral abscissa; quasi-positive matrix; asymptotic stability

1. Introduction

RNA viruses exist as heterogeneous populations of diverse genotypes in cell culture experiments[40,41], within individual hosts [1,18,22,26,29] and in populations of hosts [21,32]. The originof this diversity is in the high frequency of mutation occurring during virus replication withinthe host cell. Indeed, it has been estimated that the mutation rate of RNA viruses, defined as thenumber of misincorporations per nucleotide copied, can be as high as 10−3 [8]. Because RNA viralgenomes vary in length between 103 and 3.104, the estimates of RNA viral mutation rate lead to theconclusion that during RNA genome replication very few progeny genomes escape mutation [11].A fraction of the mutant progeny is capable of repeating the replication and diversification roundsin millions of host cells, which leads to the formation of a heterogeneous population of viralgenotypes, called quasispecies.

The concept of a viral quasispecies, defined as ‘a dynamic distribution of non-identical butclosely related mutant and recombinant viral genomes subjected to genetic variation, competitionand selection and which acts as a unit of selection’[8] is currently relatively widely adopted within

*Corresponding author. Email: tvassile@nsf.govAuthor Emails: zhou4@llnl.gov; zemla1@llnl.gov

ISSN 1751-3758 print/ISSN 1751-3766 online© 2011 Taylor & FrancisDOI: 10.1080/17513758.2010.544766http://www.informaworld.com

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the virology community ( [7,9,23,25,30,33,34,40], among many others) to qualify the variabilityof genetic sequences corresponding to the same virus species.

It is remarkable that viral quasispecies theory originated from a mathematical model of self-replicating macromolecules with non-perfect fidelity of replication by Eigen and collaborators[12,14], which was not originally meant to model virus evolution. As such, the model is quitegeneral and does not incorporate assumptions reflecting the specifics of RNA viral replication.Such specifics include the dependence of viral replication on the presence and quantity of viralpolymerase and other virus encoded non-structural and structural proteins and the possible roleof complementation and phenotypic mixing for maintaining the existence of mutants that wouldbe otherwise unable to replicate.

One of the predictions of Eigen’s model, the existence of the error threshold of the mutationrate, and specifically its application to virus evolution, has been an object of controversy. In themodel, if the mutation rate is increased, initially the equilibrium frequency of the wild type – theone with the highest fitness value – decreases and the frequencies of the mutants increase. Abovea certain value of the mutation rate – the error threshold – the original master sequence (the wildtype) loses its dominance, becoming a very small fraction of the population and the quasispeciesconsists of a large number of genotypes in low concentrations. This was interpreted as informationloss [13], and the conjecture that virus quasispecies exist near the error threshold was put forward.Assuming that there is no back mutation of the mutants to the original master sequence, Eigen andSchuster [14] calculated the approximate value of the error threshold as inversely proportionalto the sequence length. The error threshold was demonstrated in computational experiments withEigen’s model by Swetina and Schuster [37] and later repeated by other authors [15,35,44].

Several authors express skepticism regarding the value and even the existence of the errorthreshold for viral quasispecies. Iranzo and Manrubia [20] point out that the quasispecies modeldoes not take into account the large number of genotypes expressing the same phenotype; thus viralquasispecies would be expected to withstand significantly higher error thresholds [20]. Cases-Gonzales et al. [6] suggest that with real viruses, a large expansion through sequence space cannotoccur, and that the increase in error rate results in a decrease of specific infectivity, which canlead to the extinction of the population with modest expansion in sequence space. Based on asimpler model approximating Eigen’s, Summers and Litwin [36] argue that the error thresholdin Eigen’s model exists only because of the unrealistic assumption that all mutant genotypes,different from the master sequence from which they originated, are able to replicate, no matterhow many errors they contain compared to the master. Similarly, Belshaw et al. [2] consider asimple model approximating the behaviour of Eigen’s system and demonstrate that (accordingto their model) the wild type sequence would lose dominance if the mutants can replicate butwould not mutate back to the wild type. One can see easily (in this model) that if back mutationis allowed, this model [2] does not exhibit an error threshold.

It is important to note that all simulations exhibiting the error threshold ( [37] and later repeatedby other authors [15,35,37,44]) were done for single-peak fitness landscapes (where the wild typehas the highest fitness and the rest of the genotypes have equal or lower fitness). However, sincethe dynamical properties of a model may differ for different model parameters, it is plausibleto expect that if certain relations between the replication and degradation rates hold, the modelwild type may not lose its dominance no matter how high the mutation rate becomes. Indeed,Wiehe [43] considered a simplification of Eigen’s model and demonstrated that while the errorthreshold exists in a single-peak landscape, this is no longer true for other fitness landscapes.

However, simulations of the Eigen model demonstrating the error threshold [15,37,44] weredone not only just for single-peaked landscapes but also exclusively for degradation rates thatwere equal among all genotypes, including the wild type. Similarly, most of the theoretical stud-ies make the same assumption. It is reasonable to assume that well adapted virus genotypeswould be more effective in disabling cellular mechanisms of antiviral defense, including RNA

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interference mechanisms for degrading genomic and mRNA [17,27, p.195; 42]. Indeed, severalauthors reviewed in [38] provide evidence of fast virus adaptation to viral mRNA degradation byshort interfering RNA exhibited by HIV [4] and by polio virus [18]. Knipe et al. [24] find thattemperature-sensitive VSV mutant proteins were degraded at a significantly higher rate than thewild type virus; Runkler et al. [31] show that single residue substitutions at a specific position ofthe M protein of measles virus that exist only in lab conditions have a higher degradation rate thanthe wild type and a similar mutant existing in nature. While it is also possible that some mutantsmay have lower degradation rates than the wild type, in general there is no reason to expect thatall genotypes would have equal degradation rates.

Therefore, it would be instructive to explore how differing degradation rates and multi-peakfitness landscapes would affect the dominance of the wild type, and therefore, the existence of anerror threshold, in Eigen’s model. This paper provides analysis of the dependence of the equilib-rium of Eigen’s quasispecies model on the magnitude of the mutation rate with this goal in mind.We ask the question whether the wild type can remain dominant for all values of the mutationrate. To answer this question we analyse which is the highest sequence frequency when the muta-tion rate changes from 0 to 1. Mathematical analysis was possible only for small (approximately0) and large (approximately 1) values of the mutation rate. One could argue that consideringvalues of the mutation rate close to 1 lacks realism. The goal of the analysis for values of themutation rate close to 1 is to gain insight about the movement of the quasispecies equilibrium inquasispecies space when the mutation rate increases. The equilibrium is a point in quasispeciesspace that lies on a curve, parameterized by the mutation rate parameter q and connecting, aswe show, the unique stable equilibrium at q = 0 and the unique stable equilibrium at q = 1. Itis conceivable that if we find the conditions on the Eigen model parameters for which the wildtype is dominant for both low and high mutation rates, it might remain dominant for all valuesof the mutation rate and, therefore, preclude the existence of an error threshold. We do establishsuch conditions and further show, by computational experiments with Eigen’s model, that thisindeed is possible. Interestingly, our mathematical analysis shows that if the model parameterssatisfied the conditions of the computational experiments published in [37] and others, i.e. if thedegradation rates were equal and the fitness landscape had a single peak, the wild type cannotremain dominant for all values of the mutation rate. Furthermore, our analysis not only explainsthe computational observations in [37] but also derives the conditions under which, according toEigen’s model, a mutant genotype may emerge as dominant.

2. Methods, notations and definitions

The following represent a comprehensive list of notations and definitions used throughout the text.q is the mutation rate, equal to the probability of one point mutation per one sequence replication;

Sj is the binary (made of 0 and 1) ‘genotype’ sequence indexed j ; n is the number of positionsin a sequence; N is the number of all possible sequences of length n (N = 2n); d(i, j) is theHamming distance between Sj and Si , i.e. the number of positions at which Sj and Si differ.

Sequences Si and Sj are called complementary if they differ in all positions (this means thatif Si has a 0 at some position, the complementary sequence has 1 at the same position); i.e. theirHamming distance is equal to n.

Qij is the probability of mutation of Sj into Si during one sequence replication; Qii the fidelityof replication of Si ; Qii + ∑

j �=i Qij = 1. We assume uniform per nucleotide mutation rate andindependent events of point mutations. Then

Qij = (1 − q)n−d(i,j)qd(i,j). (1)

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Q = (Qij ) is the mutation probability matrix; Ai > 0 is the replication rate of Si ; A = diag(Ai),(diagonal matrix with elements Ai) is the replication rate matrix; Di is the degradation rate of Si ;D = diag(Di) is the degradation rate matrix; AiQii − Di is the fitness of Si for a given fidelityof replication Qii < 1; Fi = Ai − Di ≥ 0 is the excess productivity of type i, also here called‘nascent fitness’ (to distinguish from the fitness when replication is not perfect); F = diag(Fi) isthe nascent fitness matrix. By Im we denote the identity m × m matrix.

In the proofs of the results below we use the well known concepts of local and global stabilityof an equilibrium of a dynamical system. We use the sufficient conditions that guarantee that anequilibrium is (i) locally asymptotically stable (l.a.s.) or (ii) unstable – i.e. that the eigenvalues ofthe Jacobian of the system have, respectively (i) only negative, or (ii) at least one positive real part.To prove the existence of a continuous curve of equilibria we use the Implicit Function Theorem.In various parts of the proofs we use the following properties of quasi-positive matrices.

Definition. A square matrix M is called quasi-positive, if its off-diagonal elements arenon-negative.

In particular, if a quasi-positive matrix has only positive off-diagonal entries, it is irreducibleand has the properties described below.

Spectral property of irreducible quasi-positive matrices. If M is an irreducible quasi-positivematrix, it has a simple real eigenvalue s(M) such that s(M) > �λi(M) for all other eigenvalues λi

of M, and a positive eigenvector �ω(M) corresponding to s(M) such that any positive eigenvectorof M is a multiple of �ω. s(M) is called the spectral abscissa of M.

Majorization property of quasi-positive matrices. If M and L are quasi-positive matrices andM < L, then s(M) < s(L). Here the inequality M < L implies that each element of M is less orequal to the respective element of L and M �= L.

(M) In particular, if L has positive off-diagonal entries, then M + L is irreducible. Thus, if M

and L are quasi-positive and L has positive off-diagonal entries and M < L, then s(M) < s(L).These properties are easy corollaries of similar ones for non-negative matrices derived in [3].

2.1. The model

Eigen’s quasispecies model has the form

xi = [AiQii − Di − f (�x)]xi +∑k �=i

AkQikxk, i = 1, . . . , N, (2)

where x denotes time derivative, f (�x) = ∑(Aj − Dj)xj is the average excess productivity, also

equal to the average fitness in the absence of mutation (average nascent fitness) and the mutationprobabilities are symmetric: Qij = Qji and satisfy the relation

∑i Qij = ∑

j Qij = 1.We note that if the initial conditions satisfy

∑xi(0) = 1, then

∑i

xi(t) = 1, for all t. (3)

(Indeed, this is easily seen by summing up all equations of the model:∑

i xi = f (x)(1 − ∑i xi).

Obviously∑

i xi (0) = 0 and the same zero sum is valid for all higher derivatives of∑

i xi (0)

which implies∑

i xi (t) = 0, i.e.∑

i xi(t) = const = 1.)Because of the relation (3), xi(t) has the meaning of concentration or frequency of the genotype

sequence Si .Further, if all sequences are capable of replication, i.e. Ai > 0, for all i, and if the prob-

abilities of mutation Qij are positive, Eigen’s model (2) has a unique positive equilibrium

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�x∞ = (x∞1 , . . . , x∞

N )T (where x∞j is the frequency of Sj at equilibrium), that satisfies

[QA − D − f (�x)IN ]�x = 0. (4)

This follows from the spectral property of the irreducible quasi-positive matrix QA − D; �x∞ isthe positive vector belonging to s(QA − D) satisfying the relation f (�x∞) = s(QA − D).

It is easy to see that all solutions of system (2) with initial conditions in Rn+ remain in Rn+, i.e.the model is well posed. In all considerations further we assume that �x ∈ Rn+.

Since the excess productivities Fi ≥ 0, then s(QA − D) = f (�x) = ∑Fixi ≥ 0. This implies

that the equilibrium �x∞ is globally asymptotically stable (g.a.s.) in the positive cone RN+ , i.e. allsolutions of the model, with initial conditions in RN+ , converge to �x∞. Indeed, if s(QA − D) = 0,i.e. f ≡ 0, the system (2) is linear and can be solved directly for �x to see that �x → �x∞. Ifs(QA − D) > 0, then the substitution �y = �xe

∫ t

0 f (�x(s)) ds transforms (2) into a linear system �y =(QA − D)y. It is then easy to show (as done in [15]) that xi(t) has the form

xi(t) = x∞i es(QA−D)t + �i(t)

Bi + es(QA−D)t + �i(t),

where �e−s(QA−D)t → 0, t → ∞ and �e−s(QA−D)t → 0, t → ∞ and Bi are constants and that�x(t) → �x∞, t → ∞, i.e. �x∞ is g.a.s. in Rn+.

Since we consider mutation probabilities Qij of the form (1), then the mutation probabilitymatrix and the equilibrium are functions of the mutation rate q : Q = Q(q), �x∞

i = �x∞i (q). For

0 < q < 1, the mutation probability matrix Q(q) is a positive matrix. Therefore, �x∞(q) exists, isunique and globally stable in RN+ for all q in the open interval (0, 1) and is the solution of

[Q(q)A − D − s(Q(q)A − D)IN ]�x = 0; f (�x(q)) = s(Q(q)A − D). (5)

The Jacobian J of Equation (2) at �x∞(q) has the form

J�x∞(q) = Q(q)A − D − f (�x∞(q))IN − �(q), (6)

where � is a matrix with elements Fjx∞i (q).

In what follows, we assume without loss of generality that the nascent fitness of genotype S1

is the highest:

F1 = A1 − D1 > Fi = Ai − Di, i �= 1. (7)

3. Results

3.1. A simple model with N = 2

We first consider a case of Equation (2) with only two genotypes representing two sub-populationsof the quasispecies: one with higher nascent fitness than the other.Apart from bringing new insight,this example will be helpful in analysing a more complex model with N > 2. We assume againthat S1 has higher nascent fitness than S2:

A1 − D1 > A2 − D2. (8)

The quasispecies model then takes the form:

x1 = [A1(1 − q) − D1 − f (�x)]x1 + A2qx2 = φ1(�x, q)

x2 = A1qx1 + [A2(1 − q) − D2 − f (�x)]x2 = φ2(�x, q),(9)

where �x = (x1, x2) and f (�x) = (A1 − D1)x1 + (A2 − D2)x2.

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We shall prove the following result.

Theorem 3.1 Suppose that Equation (8) holds. Then the wild type remains dominant for all q ∈(0, 1); i.e. x1(q) > x2(q) for all q ∈ [0, 1] if and only if one of the following two conditions holds:

(a)A1 − A2 ≤ 0

or

(b)A1 − A2 > 0 and A2 + D2 ≥ A1 + D1,

(10)

Comment. Before proceeding to the proof we note that while condition (8) guarantees that whenthe fidelity of replication is perfect (i.e. the mutation rate q = 0) the sequence with the higherexcess production rate (i.e. the wild type) will be dominant (this is proved for the general case inthe next section), conditions (a) and (b), together with Equation (8), guarantee that the wild typeis dominant for very high mutation rates (q ≈ 1).

When conditions (8) and (a) are valid together, they imply that D2 > D1, i.e. the mutant isdegraded at a higher rate. When the mutation rate is very high, the replication of the mutantwould contribute mostly to production of wild type and the wild type would contribute mostly toproduction of mutant; however the mutant would produce wild type at a higher rate (A2 > A1) thanthe wild type would produce the mutant. However, this is not sufficient to ensure the dominanceof wild type at high mutation rates; it is also necessary that the mutant is degraded faster thanthe wild type.

Conditions (8) and (b) imply again that D2 > D1, i.e. the mutant is degraded at a higher rate.However, since the replication rate of the wild type is higher than the replication rate of the mutant,and thus, at high mutation rates the wild type would produce the mutant at a higher rate (A1 > A2)than the mutant would produce wild type. To guarantee that the wild type will be dominant forhigh mutation rates, it now not sufficient that D2 > D1, but it is necessary that the superiority ofthe mutant’s degradation rate over the wild type’s is higher than the superiority of the wild type’sreplication rate over the mutant’s: D2 − D1 > A1 − A2.

In conclusion, as the theorem shows, the combination of condition (8) with either (a) or (b)guarantees that the wild type remains dominant independently of how large the mutation ratebecomes, i.e. the fulfillment of these conditions precludes the existence of an error threshold.

Proof We can write the equilibrium equation as

(L − f (�x∞)I )�x∞ = 0, (11)

where

L =(

A1(1 − q) − D1 A2q

A1q A2(1 − q) − D2

). (12)

As �x∞ > 0 and because L is quasi-positive, it follows from the spectral property (above) that�x∞ > 0 is the unique eigenvector corresponding to s(L) and s(L) = f (�x∞).

s(L) = 1

2(S1(q) + S2(q)) +

√(S1(q) − S2(q)

2

)2

+ A1A2q2, (13)

where Si(q) = Ai(1 − q) − Di .

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Therefore, f (�x∞) = S1(0)x∞1 + S2(0)x∞

2 = s(L) and x∞1 + x∞

2 = 1 from which we obtain

x∞1 = s(L) − S2(0)

S1(0) − S2(0). (14)

Let us now find conditions (if any) for which x∞1 > x∞

2 , i.e. x1 > 12 , for all q ∈ (0, 1). Using

Equations (13) and (14) after simple but tedious algebraic calculations we obtain

x1 > 12 ⇐⇒ [(A1 − A2) − (D1 − D2)][(A1 − A2) − (D1 − D2) − 2q(A1 − A2)] > 0. (15)

Because of Equation (8), the latter inequality is possible for all q ∈ (0, 1) only if (A1 − A2) −(D1 − D2) − 2q(A1 − A2) > 0.

(a) If A1 − A2 ≤ 0, the latter is always true.(b) If A1 − A2 > 0, the inequality is possible only if for all q ∈ (0, 1), (D1 − D2)/(A1 − A2) <

1 − 2q, which is valid if and only if (D1 − D2)/(A1 − A2) < −1 which is equivalent toA1 − A2 > 0 and A1 + D1 < A2 + D2.

Therefore, the wild type would remain dominant for all values of q ∈ (0, 1) if and only ifA1 − A2 ≤ 0 or A1 − A2 > 0 and A1 + D1 ≤ A2 + D2. �

3.2. General case

3.2.1. Analysis for q ≈ 0

(a) In the absence of mutation (q = 0) the wild type is the only one that persists.If q = 0, Qij = 0, i �= j and the system can be written as

�x = [F − f (�x)IN ]�x, i = 1, . . . , N. (16)

Theorem 3.2 If F1 > Fi, i = 2, . . . , N the equilibrium (1, 0, . . . , 0)T of Equation (16) is g.a.s.in Rn+.

Proof Obviously (1, 0, . . . , 0)T is an eigenvector of the eigenvalue F1 and an equilibrium ofEquation (2). All other equilibria are unstable because the Jacobian at each such equilibriumalways has the eigenvalue F1 − ∑N

i=2 Fixi > 0.

Further, system (16) is equivalent to xi(t) = xi(0)eFi t−∫ t

0 f (�x(s)) ds . Using this equality, and thefollowing one:

d

dte∫ t

0 f ds =∑

i

Fixi(0)eFi t ,

we solve for f :

f (�x(t)) =∑

Fixi(0)eFi t∑Fixi(0)

∫ t

0 eFis ds

and take the limit at t → ∞ to obtain

limt→∞ f (�x(t)) = F1.

Therefore, it follows from Equation (16) that xi(t) → 0 for all i �= 1 and because ofEquation (3), it follows that limt→∞ x1 → 1. �

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538 T. Kostova et al.

(b) For q > 0 but close to 0, the equilibrium lies on a smooth curve starting at q = 0 from(1, 0, . . . , 0)T. For small values of q the wild type frequency x∞

1 decreases but remains largerthan the frequencies of the remaining types which increase.

Theorem 3.3 For q in some interval I0 = (0, ε0), there exists a smooth (of class C∞) RN

curve C0 parameterized by q, such that for q ∈ I0, the unique positive equilibrium �x∞(q) of thesystem (16) lies on C0 and such that limq→0 �x∞(q) = (1, 0, . . . , 0). In this interval the wild typefrequency x∞

1 decreases and the frequencies of the remaining types decrease.

Proof The existence of a smooth (of class C∞) RN function of q in an interval around 0 is easilyestablished by the Implicit Function Theorem, keeping in mind that the Jacobian is non-singularat (1, 0, . . . , 0) and that Qij (q) are polynomials of q, i.e. infinitely differentiable. However, weneed to verify that x∞

i (q) ∈ (0, 1). This is easy to see by finding the derivatives of x∞i at q = 0.

Specifically, we find that

dx∞1

dq(0) = 1

F1

⎛⎝−NA1 − A1

∑j :d(1,j)=1

Fj

F1 − Fj

⎞⎠ < 0. (17)

If d(1, j) = 1, i.e. the sequence with index j is only one mutation away from the wild type:

dx∞j

dq(0) = A1

F1 − Fj

> 0 (18)

For the sequences that are two mutations away from the wild type, i.e. d(1, j) = 2:

d2x∞j

dq2(0) = 2A1

F1 − Fj

> 0 (19)

while dx∞j /dq(0) = 0, and in general, if d(1, j) = k, k = 1, . . . , N :

dkx∞j

dq2(0) = k! A1

F1 − Fj

> 0 (20)

while all lower derivatives are equal to 0.From the above inequalities we see that for small values of q, x∞

1 decreases when q increases(dx1/dq(0) < 0) and for j > 1, x∞

j (q) are increasing functions of q: x∞j ≈ dkx∞

j /dqk(0)qk ,where k = d(1, j). Therefore, for small q > 0 the continuation �x∞(q) of �x∞(0) ∈ (0, 1)N andthus coincides with the unique positive equilibrium of Equation (16).

Obviously, as x∞j are continuous, it follows that in some interval q ∈ (0, qmin), x1 > xi ,

i = 2, . . . , N . �

3.3. Analysis for q ≈ 1

(a) For q = 1 model (2) has a l.a.s. non-negative equilibrium �ζ which is an eigenvector to thespectral abscissa. When q = 1, Qik = 1 if d(i, k) = n (i.e. Si and Sk are complementary) andQik = 0 otherwise. We rearrange the sequences Si in the following way: the index of the wildtype (with the largest nascent fitness) is 1 and the index of its complement is 2; next, we pick anysequence different from these two and give it an index 3 and give its complement sequence index

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4, and continue in the same way, pairing complementary sequences. Then Q(1)A − D has theblock diagonal form

Q(1)A − D =

⎛⎜⎜⎜⎜⎝

L1 0 . . . . . . 00 L2 0 . . . 00 0 L3 . . . 0. . . . . . . . . . . . . . .

0 0 . . . . . . LN/2

⎞⎟⎟⎟⎟⎠ , (21)

where

Li =(−D2i−1 A2i

A2i−1 −D2i

)(22)

and 0 is a two-dimensional square zero matrix.Note that since for each i, Li are quasi-positive, they have two real eigenvalues, one of

which is s(Li). Let k0 be the index for which Lk0 has the largest spectral abscissa: s(Lk0) =maxi=1,...,N/2 s(Li). Obviously s(QA − D) = s(Lk0).

Theorem 3.4 Let q = 1. Then QA − D has a non-negative eigenvector belonging to s(QA − D)

which is an l.a.s. equilibrium of Equation (2).

Proof We first consider the case when s(QA − D) has multiplicity 1, i.e. we assume that

s(Lk0) = maxi=1,...,N/2

s(Li) > s(Lj ), j �= k0. (23)

In this case QA − D has a unique eigenvector �ζ such that ζi = 0, for i �= 2k0 − 1 or 2k0 andζ2k0−1, ζ2k0 solve the system

Lk0

(ζ2k0−1

ζ2k0

)=

(00

). (24)

Here(

ζ2k0−1

ζ2k0

)is the positive eigenvector of Lk0 , corresponding to s(Lk0) such that ζ2k0−1 + ζ2k0 = 1.

We also recall that f (�ζ ) = s(Lk0).The Jacobian (6) at �ζ is a block diagonal matrix J�ζ = diag(J1, . . . , JN/2) where

Jk0 =(−D2k0−1 − F2k0−1ζ2k0−1 − f (�ζ ) A2k0 − F2k0ζ

∞2k0−1

A2k0−1 − F2k0−1ζ∞2k0

−D2k0 − F2k0ζ∞2k0

− f (�ζ )

). (25)

and

Ji =(−D2i−1 − f (�ζ ) A2i

A2i−1 −D2i − f (�ζ )

)for i �= k0. (26)

Since ζ∞2k0−1 < 1, ζ∞

2k0< 1, then A2k0 − F2k0ζ

∞2k0−1 > 0 and A2k0−1 − F2k0−1ζ

∞2k0

> 0. There-fore, Jk0 is a quasi-positive matrix and Jk0 < Lk0 − f (ζ )I2. Therefore s(Jk0) < s(Lk0 −f (ζ )I2) = 0.

Further, for i �= k0, s(Ji) = s(Li) − f (ζ )I2 < 0. This proves the local asymptotic stabilityof ζ .

In the case when s(QA − D) has multiplicity m > 1, let Lkj, j = 1, . . . , m be those for which

s(Lkj) = s(QA − D) and let (z2kj −1, z2kj

) be their positive 2 × 2 eigenvectors. Then the vector �ζwith coordinates �ζ2kj −1 = z2kj −1, ζ2kj

= z2kj, j = 1, . . . , m and zeros otherwise is an eigenvector

of QA − D. Using a method similar to the above, we can see that the Jacobian at �ζ Equation (6)satisfies s(J�ζ ) < 0, i.e. �ζ is l.a.s. �

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(b) For q < 1 but close to 1, the positive equilibrium of Equation (2) lies on a smooth curvestarting at q = 1 from the unique l.a.s. equilibrium �ζ .

Theorem 3.5 For q in some interval I1 = (1 − ε, 1), there exists a smooth (of class C∞) RN

curve C1 parameterized by q, such that for q ∈ I1, the unique positive equilibrium �x∞(q) of thesystem (16) lies on C1 and limq→1 �x∞(q) = �ζ .

Proof The existence of C1 is guaranteed by the Implicit Function Theorem because the eigen-values of the Jacobian J�ζ of Equation (2) at �ζ lie in the left half of the complex plane, i.e. theJacobian is non-singular. The C∞ smoothness follows from the fact that Qij are polynomials of q,i.e. infinitely differentiable. Let us assume that the curve is defined by the RN function ξ(q). Weneed only show that ξ∞

i (q) ∈ (0, 1) for q sufficiently close to 1. Because the non-zero coordinatesof �ζ are between 0 and 1, we need only show that if ζi = 0, then ξ∞

i (q) > 0 for q ≈ 1. We showthis by finding the derivatives of ξi with respect to q.

Without loss of generality, let ξi = 0, for i = 2l − 1 for some l. Then ξ2l = 0 as well since thisis the frequency of the complementary sequence with index 2l − 1. Differentiating the right-handside of Equation (2) with respect to q, we obtain for p = 2l − 1 and p = 2l:

∑j

Aj

dQpj

dqξj +

∑j �=p

(AjQpj − Fjξp)dξj

dq− (Di + f (�ξ))

dξp

dq. (27)

We next note that for q = 1:Qp,j = 1, if d(p, j) = n and 0 otherwise, and

dQp,j

dq= n, if d(p, j) = n

− 1, if d(p, j) = n − 1 (28)

0, otherwise.

Using these, we get the following equations for dξi/dq :

(Ll − f (�ξ)I2)

(dξ2l−1

dq,

dξ2l

dq

)T

= −∑

j

Aj

dQpj

dqξj . (29)

The sum at the right is either 0 or less than 0, when keeping in mind Equation (28) and the formog �ζ . If it is less than zero, we note that the matrix −(Ll − f (�ξ)I2) is an M-matrix and thus, itsinverse is positive.It follows that (dξ2l−1/dq, dξ2l/dq)T is a negative vector.

If the sum at the right in Equation (29) is zero, we consider the second and further derivatives,as was done for the case q = 0 in the previous sub-section. We find that eventually the higherorder derivatives (dξ2l−1/dq, dξ2l/dq)T are negative. �

3.4. For q ∈ [0, 1] the positive equilibria lie on a smooth curve connecting (1, . . . , 0)T andthe l.a.s. equilibrium ζ

Theorem 3.6 �x∞(q) defines a smooth curve (of class C∞) C in RN .

Proof �x∞(q) is the eigenvector of Q(q)A − D belonging to the spectral abscissa s(Q(q)

A − D) = s∗(q). Note that for q ∈ (0, 1), Q(q) is a positive matrix, i.e. QA − D is irreducibleand quasi-positive.

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(a) We first note that since s∗(q) is a simple root of the characteristic polynomial det |Q(q)A −D| = 0, it is a smooth (of class C∞) function of q. (This follows easily from the ImplicitFunction Theorem and is a well known result.)

(b) Further, let us represent �x∞(q) as

�x∞(q) =(�x∞

n−1xn

)

where �x∞n−1 = (x∞

1 , . . . , x∞n−1)

T and QA − D as

Q(q)A − D =⎛⎝ Q1nAn

B(q) . . .

Qn1A1 . . . QnnAn − Dn

⎞⎠

where B is an n − 1 × n − 1 quasi-positive matrix.

Therefore, if B(q) − s∗(q)In−1 was invertible, �x∞ would be a smooth (C∞) function of q

determined from

�x∞n−1 = (B(q) − s∗(q)In−1)

−1(−An)

⎛⎝ Q1n

. . .

Qn−1n

⎞⎠ x∞

n

n∑i=1

xi = 1.

(30)

To complete the proof we will show that B(q) − s∗(q)In−1 is invertible. Let

Z(q) =⎛⎝ 0

B(q) 00 . . . −D

⎞⎠ . (31)

Obviously Z(q) is a quasi-positive matrix, Z(q) < Q(q)A − D and Z + QA − D is anirreducible quasi-positive matrix. From the Majorization property (Section 2) it follows thats(Z) = max(−D, s(B(q))) < s(Q(q)A − D). Therefore

s(B(q)) < s(Q(q)A − D) = s∗(q).

Therefore, B(q) − s∗(q)In−1 is invertible. �

We have also proved that for q ≈ 0 and for q ≈ 1 there exist curves C0 and C1 starting,respectively, from the l.a.s. equilibria (1, 0, . . . , 0)T, for q = 0 and ζ , for q = 1 and containingthe positive equilibria.As we know, for each q ∈ (0, 1), �x∞(q) is unique. Therefore, for q ∈ (0, 1)

C0 and C1 coincide with C. Thus, C has endpoints at (1, 0, . . . , 0)T, for q = 0 and ζ , for q = 1.

3.5. Explicit relations between replication and degradation rates to infer the dominantsequence at high mutation rates

If we assume that s(QA − D) = s(Lk0) is a simple eigenvalue, then for q = 1, the larger of thetwo positive coordinates ζ2k0−1 or ζ2k0 is the dominant frequency. If k0 = 1, then either the wildtype or its complement is dominant at q = 1. Using Theorem 3.1 we can discriminate betweenthese two possibilities. This implies the following result.

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Theorem 3.7 Let the sequences comprising the quasispecies be indexed as described in thebeginning of Section 3.3 and the conditions, used throughout the paper

(a) Ai > 0, Di > 0, i = 1, . . . , N;(b) F1 = A1 − D1 > Ai − Di > 0, i = 1, . . . , N

(32)

be satisfied.

(i) If the following condition holds,

(c)s(L1) � − 12 (D1 + D2) + 1

2

√(D1 − D2)2 + 4A1A2

> − 12 (D2i−1 + D2i ) + 1

2

√(D2i−1 − D2i )2 + 4A2i−1A2i

� s(Li), i = 2, . . . , N/2,

(33)

then the wild type is dominant for high mutation rates if and only if

(d) A1 − A2 ≤ 0 or A1 − A2 > 0 and A1 + D1 < A2 + D2. (34)

(ii) If (c) is true, but instead of (d), its negation is true: A1 − A2 > 0 and A1 + D1 > A2 + D2,

the complement sequence of the wild type is dominant for high mutation rates.(iii) If (c) is not fulfilled, neither the wild type nor its complement will be dominant for high

mutation rates (q ≈ 1). Then for the index k0 > 1 satisfying (23), one of the sequencesS2k0−1 or S2k0 will be dominant for high mutation rates. Namely, if, for example, A2k0−1 −D2k0−1 > A2K0 − D2k0 , and either A2k0−1 ≤ A2k0 or A2k0−1 > A2k0 and A2k0−1 + D2k0−1 <

A2k0 + D2k0 , then S2k0−1 is dominant for q ≈ 1.

Comment.When the mutation rate is very high, each sequence mutates with high probability intoits complementary sequence and with very low probability into any other sequence. As seen fromthe analysis in Theorem 3.4, at very high mutation rates the couple sequence S2l−1 – complementS2l , which has the highest average fitness (A2k0−1 − D2k0−1)ζ

∞2k0−1 + (A2k0 − D2k0ζ

∞2k0

) = s(Lk0)

wins the competition and dominates the quasispecies. Thus, condition (c) guarantees that at veryhigh mutation rates the couple of the wild type and its complement dominates the quasispeciespopulation if it has the highest average fitness. In addition, at high mutation rates, the wild typewill dominate its complement if and only if (see comments after Theorem 3.1) the degradationrate of the complement is sufficiently higher than the wild type’s one.

Note. If Di = D (uniform degradation rates) and A1 > Ai, i > 1 (single-peak fitness land-scape), as assumed in [15,37,44], the model satisfies the conditions (ii) of Theorem 3.7.

3.6. Necessary conditions for the dominance of the wild type

Obviously, the validity of (a), (b), (c) and (d) implies that the curve of equilibria �x∞(q) ends at apoint (ζ1, ζ2, 0, . . . , 0)T with ζ1 > ζ2. If we consider separately the curves x∞

i (q), q ∈ [0, 1], thefulfillment of (a)–(d) is a necessary (but not sufficient) condition for the curve x1(q) to be locatedentirely above any of the other curves xi(q), i.e. for the frequency of the wild type to be higherthan any other frequencies, for all q, i.e. the wild type to retain its dominance independently ofthe value of the mutation rate. If (c) or (d) do not hold, the wild type will lose dominance for somevalue of the mutation rate when it increases from 0. Conditions (a)–(d) could also be sufficientfor dominance of the wild type throughout the mutation rate range in some cases. Indeed, inExamples 1 and 2 below, we demonstrate that the fulfillment of (a)–(d) can guarantee dominanceof the wild type for all mutation rate values. Further, Example 3 demonstrates the emergence ofa new dominant type because of the violation of condition (c).

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3.7. Examples

We constructed examples to demonstrate the preservation of the wild type dominance over thefull range of variation of the mutation rate from 0 to 1, as well as cases when the wild type losesdominance, as predicted by the theory developed in the previous section. The quasispecies model

Table 1. Parameters used in the numerical examples.

Example n N Values of Ai, i = 1, . . . , N Values of Di, i = 1, . . . , N

1 2 4 2, 1.4, 1.5, 1.8 0.1, 1.3, 1.3, 1.52 4 16 2, 0.5, 1.9, 1.8, 1.8, 1.7, 1.9, 1.7, 1.7,

1.65, 1.95, 1.9, 1.65, 1.8, 1.8, 1.91, 0.4, 1.8, 1.75, 1.7, 1.6, 1.85, 1.6, 1.6,

1.6, 1.85, 1.85, 1.6, 1.75, 1, 13 4 16 2, 0.05, 1.9, 1.8, 1.8, 1.7, 1.9, 1.7, 1.7,

1.65, 1.95, 1.9, 1.65, 1.8, 1.8, 1.91, 0.01, 1.8, 1.75, 1.7, 1.6, 1.85, 1.6, 1.6,

1.6, 1.85, 1.85, 1.6, 1.75, 1, 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Mutation rate q

Equ

ilibr

ium

con

cent

ratio

ns

Fittest species maintains dominance.

Figure 1. Equilibrium concentrations calculated for Example 1. ‘�’ → x1, ‘ ∗ ’ → x2, ‘ + ’ → x3, ‘ · ’ → x4.

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544 T. Kostova et al.

with mutation probability matrix Q as defined in Section 2 and parameter values given belowwas solved using Matlab on a sufficiently long time interval so that at the end of the interval thesolution was in the vicinity of the equilibrium point and did not change up to the 16th digit. Thevalue of the mutation rate q was varied from 0.01 to 0.99 and the equilibrium concentrations foreach q were plotted. The parameters of the three examples are presented in Table 1.

In Examples 1 and 2 the parameters were chosen so that the conditions (a)–(d) of Theorem 3.7are fulfilled. Indeed, in Example 1, conditions (a), (b) and (d) are obviously fulfilled and s(L1) =1.0776 and s(L2) = 0.2462. Based on Theorem 3.7, it is expected that for Example 1, x∞

1 (q) >

x∞2 (q) and x3(q) ≈ 0, x4(q) ≈ 0 for q ≈ 1, i.e. x∞

1 (q) > x∞i (q), i = 2, 3, 4.As seen on Figure 1,

not only is this true, but the wild type remains dominant for all values of the mutation rate q.Similarly, in Example 2, (a), (b) and (d) hold and s(L1) is the largest spectral abscissa and notonly it is true that x∞

1 (q) > x∞i (q), i = 2, . . . , 16 for q ≈ 1 but this relation holds for all values

of q ∈ (0, 1) as is seen on Figure 2.In Example 3 we chose the parameters so that condition (c) of Theorem 3.7 is not fulfilled.

Specifically, s(L1) = 0.0824 while s(L8) is the largest of all s(Li) and equal to 0.8493. As

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Mutation rate q

Qua

sisp

ecie

s eq

uilib

rium

freq

uenc

ies

Quasispecies equilibrium frequencies versus mutation rate.The fittest of 16 types preserves dominance.

Figure 2. Equilibrium concentrations calculated for Example 2. ‘�’ → x1, ‘ ∗ ’ → x2, the complement of the wild type.The wild type remains dominant for all values of q. All other 14 frequencies are represented with various symbols thatoverlap on the plot.

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Journal of Biological Dynamics 545

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Quasispecies equilibrium frequencies versusmutation rate: lowest fitness species dominant for high q.

Mutation rate q

Qua

sisp

ecie

s eq

uilib

rium

freq

uenc

ies

Figure 3. Equilibrium concentrations calculated for Example 3. ‘�’ → x1, ‘ ∗ ’ → x2, the complement of the wild type,‘ ◦ ’ → x15, ‘ � ’ → x16. The wild type loses dominance for all values of q. All other 14 frequencies are represented withvarious symbols that overlap on the plot.

expected from Theorem 3.7, S15 and S16 are the dominant sequences for q ≈ 1. (Figure 3). Thewild type (depicted by triangle symbols) loses dominance for q ≈ 0.15 and never regains it.

4. Discussion

The role of the degradation rates in the quasispecies model has been neglected in the literature.As mentioned in Section 1, the usual assumption has been that the degradation rates of all modelgenotypes are equal. However, this assumption implies that the relation between the fitnesses ofany two sequences remains the same when the mutation rate changes. Indeed, if the nascent fitness(or, excess production rate) F1 of sequence S1 (the wild type) is higher than the nascent fitness F2 ofanother sequence S2 (the mutant), but their degradation rates are equal: (A1 − D > A2 − D), thissame relation is preserved for any value of the mutation rate: A1Q11 − D > A2Q22 − D (sinceaccording to the model, the fidelity of replication has the same dependency on the mutation ratefor all sequences: Qii = (1 − q)n). In addition, such an assumption also implies that genotypes

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with higher fitness also have higher replication rate: A1 − D > A2 − D ≡ A1 > A2 . However,if the rates of degradation of the wild type and the mutant differ, these implications may no longerbe valid. First, the fact that the wild type has higher nascent fitness no longer implies that it hasa higher replication rate. Second, since an increase in the mutation rate leads to a decrease in thefidelity of replication but has no effect on the rate of degradation, the fitness of the wild type maybecome lower than the fitness of the mutant when the fidelity of replication goes down. Thus, theconcept of fitness is dependent on the value of the mutation rate. To differentiate between the casewith no mutation and the one with mutation, we introduced the term nascent fitness.

Our study demonstrates that, according to the quasispecies model, when all genotype degrada-tion rates are assumed equal, the wild type always loses dominance, since this assumption leadsto the violation of condition (d) of Theorem 3.7. Specifically, for a single-peak fitness landscape(A1 > Ai, i �= 1), conditions (a)–(c) of Theorem 3.7 are valid but condition (d) is violated andthe wild types loses its dominance to its complement for some value of the mutation rate q.Further, for equal degradation rates and a multiple-peak fitness landscape, any model genotypeSk, k = 2i or 2i − 1 satisfying the relation

√A1A2 <

√A2iA2i−1 will eventually become dom-

inant above some value of the mutation rate. Finally, if the degradation rate of the wild S1 typeis sufficiently higher than the degradation rate of another sequence Si , so that condition (c) isviolated, Si will eventually become the dominant sequence. However, if the degradation rate ofthe wild type is sufficiently lower that the degradation rates of all other sequences so that (c) and(d) hold, the wild type might remain dominant for all values of the mutation rate, as exemplified inExamples 1 and 2.

We note that the use of the terms model genotype or sequence emphasizes the fact that theresults and conclusions refer to the outcomes of Eigen’s model. Indeed, the existence and evendominance of the complementary sequence of a given sequence is improbable in the case of aviral quasispecies.

Our computational examples based on the theoretical analysis for the extreme values of themutation rate demonstrate the possibility that the error threshold may not exist when the degra-dation rates are not uniformly equal. Notably, if the degradation rate of the wild type is muchsmaller than the degradation rates of the mutants, it is easy to formulate examples where the wildtype retains its dominance. This is yet another piece of evidence that, apart from the referencesmentioned in the Introduction, that the error threshold may not necessarily exist.

Viral quasispecies possess enormous plasticity that may enable them to avoid phenomena likethe error catastrophe. Experiments with ribavirin that were meant to demonstrate the existenceof the error threshold in poliovirus [7] showed that the increase in mutation rate caused by thedrug was accompanied by a steep decrease in infectivity. This was interpreted as the virus havingpassed the error threshold. However it was later shown that poliovirus developed resistance toribavirin enabled by the emergence of a genotype coding for a polymerase with increased fidelity,which narrowed down the quasispecies diversity [28,39]. The propensity for poliovirus to adaptand avoid error catastrophe cannot be predicted by the quasispecies model because it does takeinto account the complexity of the virus replication mechanism.

We note that the error threshold as a concept is not well understood and is frequently mistakenfor the extinction threshold, both in theory [20,35,36,42] and experiment [10,39]. The conceptualdifference between error threshold and extinction threshold was made clear in [5] and recentlyreiterated by Holmes [19]. Extinction threshold is a term used to explain an abrupt change indensity or number of a population brought on by changes in important parameters, such as habitatloss or increased death rate due to various factors (e.g. human factors, disease). Whereas Eigen’smodel can demonstrate the existence of an error threshold, it does not support the existence ofan extinction threshold. The mathematical structure of Eigen’s model is such that for each set ofparameters there is only one equilibrium, which is strictly positive and globally stable. This meansthat independently of the initial genotype frequencies, the quasispecies population converges with

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time to an equilibrium in which each of the genotypes is present. Therefore, in Eigen’s model thequasispecies does not have an extinction threshold.

In conclusion, we believe that while indeed, RNA viruses exist as a quasispecies (as defined inSection 1), viral quasispecies theory is in need of further development. Viral quasispecies modelsshould include more realism, both in the representation of the genome sequences (note that in thecurrent paper we have restricted the analysis to binary sequences) and in reflecting the intricatemechanisms of viral replication and adaptation.

Acknowledgements

This work was performed partially under the auspices of the US Department of Energy by Lawrence Livermore NationalLaboratory under Contract DE-AC52-07NA27344. The work was supported by an LLNL–LLNS internally funded grantto T.K. and C.Z. through the Laboratory Directed Research and Development program. T.K.’s research was also fundedby NSF. Any opinion, finding, and conclusions or recommendations expressed in this material are those of the author anddo not necessarily reflect the views of the National Science Foundation.

The authors express gratitude to Professor Raul Andino (UCSF) for many insightful discussions of the quasispeciesphenomenon, which helped in understanding the limitations of the quasispecies model.

Last but not least, the authors are deeply thankful for the thoughtful criticisms of an anonymous reviewer whichsignificantly improved the paper.

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